Discussions, incl. M Wright, FW Lawvere & A Kock on objects & subobjects (contd.)

0:00 ...that we can't know all about hardware. We should make some steps to unfreeze this wall of ignorance. We should make some steps to be deprived of it.

7:30 It's not difficult, no. Why do you want to learn synthetic differential geometry? I mean, I think the point reflects on your point. It reflects on the people in Israel. It reflects on... He did say that the differential geomorphism had given up. He was saying that differential geomorphism in Urbana had given up. But of course the implication was that... I'm being quoted by a differential geologist in Berkeley by the name of Weinstein.

10:00 The problem is people are not interested in thinking deeply and seriously about that. Apart from thinking, you know, the work on statistics and geometry is so... It's very important for understanding the relationship between the category of sets and the category of spaces and other models, I mean, and so, as you were making the point so well in Cambridge last year, it's of tremendous importance conceptually in understanding the direction of fit of the Strachan Foundation and whether, you know, where, you know, encountering Platonist attitudes towards... Mathematical structure, much else besides. Apart from anything else, I mean, a more careful attention to synthetic differential geometry would certainly... Still defences against the sort of nonsense that's thought about black holes, as you pointed out very well. Understanding the relationship between piano arithmetic and the continuum. Anyway, what that young guy was saying about SVGs was really just in effect a criticism of his own university. Yes, I see it more as a... What he was saying about that you have to learn the Python to see science. Well, I think he's completely wrong about this division. I think what Bill said was right, and several other speakers made the point, but this definitive division between applied and pure categories is a bogus division. But should you have the time, Anders, do you actually have a point? I was very disappointed, to be honest, that nobody...

12:30 I think it might actually have helped to direct the discussion and make it more effective if Saundersman had actually stuck his neck out and decided to set the agenda of the questions that he actually posed at the end of his first talk on the first morning, to remember his hypotheses as to the seven or eight coming collisions or interactions between category theory and other fields. Um, I think it might have been... I think that would have gone into technicalities, really. Well, some of it today went, anyway, too far. Well, it just seemed to center so entirely on the international three categories here in computer science. Oh, that was a very well-organized... Well, surprise, surprise. I would, of course, like to have heard. Somebody raised the question of the interaction between category theory and physics, which he touched on on the second point. It was touched on briefly when he talked about quantum groups, theory of topologies and quantum groups, but I mean that's very abstract, I mean that's really only vestigially anything to do with physics, it's, you know, what if anything has quantum groups to do with physics, exactly. There is some connection between physics and mathematical physics. Even if the problem is about the technical mechanics, what do you think about the Ising model? The what model? The Ising model. The Ising model. The Ising model. The Ising model. The Ising model.

15:00 I-S-I-N-G is a lattice rock. There is a connection with mathematics, but it's not something you're going to apply tomorrow or in the years to come. Well, I had in mind a much more schematic connection, the sort of questions that... Jerry's interest in projectiles raises over, about whether one might have, I suppose I guess it's more about connections with logic and fields work than it is with physics, because I had in mind the example of quantum logic because of the way that one I have to think about functoriality. I mean, this construction of the area fails to be functorial at the first variable because of the adjoining conditions, the additional structure has been imposed on it to make it into a closed category, which appears to be motivated. Maybe it's only really from projective geometry, but he actually came to it from... No from quantum theory. No, I think Bill would say, well, that's because we're not in the right category. We ought to be looking at the set of endo maps into a fixed truth value object. We ought to be looking at more general category, category of all Hilbert spaces, or some further category. I mean, just as you were saying at the Cambridge meeting. But I think that, and this is the point where, I mean, work which is interesting in its own right,

17:30 from the point of view of generalization of... Closed categories to something like project art does make contact with foundational questions about how one thinks about extensionality and relative uniform separability and factorizability. I didn't really get the point. No, no, I'm sorry. Well, it's not been well expressed, I'm afraid. Just there is a motivation from physics, a direct motivation from physics in Jerry's work on this, which you were talking about with him earlier on, for this construction that he calls the projectile. And that particular construction also seems to raise interesting questions about the way that Separation of sub-objects and extensionality and the extent to which certain classical properties, classical ways of thinking of the domain are built into the topos axioms, not just in the case, obviously, of the category of sets, but more generally, the way that, I mean, you obviously, well, I think you actually refereed his paper, didn't you, for the publication? I guess it was you, I think. Well, you're welcome, sir. I wouldn't dream of saying that, I wouldn't say, but... I mean, one of the... no, sort of the ethics of the professor... The... I mean, the reason the projectile fails to be a closed category is that you don't actually have a... Am I not right? It fails to be a closed category because it's not functorial in the first variable. There's not actually a unit in the tensor product. When you impose that condition, you then recover the hiding... well, actually, it then becomes a hiding algebra. Now, the motivation behind that, that this is in some sense a kind of logic of directions, something slightly that's a further generalization, still preserves jointness, but it's a further generalization in the way that the Heiting algebra itself was a generalization in the context of variable sets of Boolean algebras.

20:00 That motivation seems to me to come, well, indirectly, not so very indirectly, from physics, or at any rate from the kind of intersection of logic and physics. That was the kind of interaction I had in mind. You know, that was the kind of, collision is perhaps the wrong word, but that was the kind of cross-fertilization of perspectives from different fields that I had in mind. In category theory, you can have a little bit of a laugh at this time. Uh, let's go to the development of a new field of the education of everybody. It's part of the new field, right? There's a new big issue. A lot of discussion. What is the issue? No, no one said anything about the new field. Who informed you about that? Who informed you about that? Thank you for your attention. I'm not a fan of computer science, but you know, for example, this journal, mathematical structure and computer science, I think there's an awful number of getting curious. And I wonder, was this journal... Specifically created by a group of getting used to the show of life.

22:30 Sorry. Sure. What can I get you? Sorry. Two beers? Two beers. Three? For you. Okay. Okay. Three beers. Very well said. Don't bother eating. This is your strike here. Who's behind this journal? The Commodity? Cambridge University Press. But they depend on their academics. And they are in favor of this. Yes, but there must be some computer scientists. Well, I'll try to correct it. I would have thought that the cross-fertilization of logic and category theory is a more clear example, but you couldn't really. So how much, what's the cost of all this? Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Uno. Solo uno. Actually, I was getting you one. I don't pay. All right. Sete miles. Sete miles. No, it's okay. I'm complicating. You're complicating matters. You're complicating. It's only another mile. And you just have another 1,000, by any chance, because otherwise I have to break 100,000, which is a... No, no, just 1,000.

25:00 That's right, there we are. That's it. There we are. Sorry, you said 7,000, no? Oh, 7,000, sorry, 7,000, another 1,000 as well. Yes, sorry, 7,000. Sorry, I've only given him 5. That's 2, isn't it? Thank you. Get Leonardo back. Thank you for watching. Do you notice that they've changed all the seats since yesterday? Maybe their insurance company got them? Rather an extraordinary coincidence, isn't it? Well, it does seem a striking coincidence. Thank you for watching. You're on that side, aren't you? Yeah, let's grab these, probably right now. Been sitting all evening anyway. By the way, Anders, you know that John Bell is still, um...

27:30 Trying to get together the contributions from the people who were at the Cambridge meeting last year with a view to publishing a book and I haven't spoken to him for about three or four months because he's in Canada now yes that's right he's gone to Toronto but Michael Redhead spoke to Cambridge University Press before he went and I think he was obviously hoping that Bill would write up the talks that he gave there for publication, which he said that he was willing to do, and he's got so many other projects on at the same time. But would you be interested in writing something, expanding on your splendid knockabout piece? Maybe. I don't know what else really sort of... Yeah, the contributions were very, very different. ...kinds of contribution. I mean, they were... You might want to write something else, of course. Oh, well, that would... Yeah. I don't know if you have any thoughts about foundations that you'd like to... Or whether you feel that you should be as old and as, you know, as revered and as... ...as revered and as old as Matej before you get around to writing anything about foundations and mathematics, maybe. You've probably got far too much work on at the moment. Yeah, they're not really suitable for conference proceedings, no. Well, would you feel it's all inclined to write something a little bit more... a little less polemical and a little bit more considered about foundations?

30:00 A lot of people have used, I like John Bell, his art on infinitesimals, have lent very heavily on your work on synthetic differential geometry. I don't know whether you've always felt that they've interpreted your own... Point of view. I mean, your point of view on the older ideas and fundamentals correctly, or whether... I'm really having nothing to say about mathematical mathematics, which I haven't learned from Luir. That's fair enough. So, in that sense, I'm... Well, it's a question, of course, of getting Bill to say what his position is in the foundation of mathematics. I know implicitly he says that all the time whenever he talks mathematics, but presenting it for an audience of philosophers, presenting it for the kind of audience that that conference was intended for, is a different thing. And I know he's almost got enormous pressures on his time. Actually, shall we take our chairs and go over there and join them. They're obviously not going... We have to wait with this bag here. Oh, right. Unless you see André... I think he's just in the loo. Yeah, probably, yes. Well, right, okay. Yeah, that's right. I was going to take his beer and his bag, but if, um, okay. Well, I'll wait with you in a minute.

32:30 No, I'm a little irritated with this discussion that John Gray comes up with. Yeah, I realized what your author said at that time. And it's not because of... not at all with his statement about VSTG. That's his and Urbana's. That's not the point, but his general statement's about that category theorists should take... I must admit I thought it was slightly ironic because we've been told for the first three quarters of the meeting about the reason that category theory wasn't held in sufficiently high refute by the mathematical community as a whole was because it was too general and too accessible because, as Sammy Eilenberg put it, the distance from the definitions to the... The frontier of research is not as sufficiently great by comparison with areas like number theory, on the one hand, and then, you know, Gray stands up at the end and says, well, we're making it much too difficult for these poor differential geometers to understand, they can't, so you must make things simpler and more accessible, you really can't have it both ways. And what does he mean by that? Do you know that from his own... It's a talk where the concepts are not steep, but quite a good deal ad hoc. Yes, very. And that's an extraordinary little table of equivalences between logic, arithmetic and relations, which really wasn't brought up. In fact, I didn't think he really made use of category theory at all in his talk. No, that is what he's making a point of, it seems, from his contribution to this night's discussion. No, now I think I know how to summarize what he was saying in plain words. We should completely strip ourselves of any of the concepts and culture we learned and completely, through several years of hard study, assimilate the culture of the computer scientists in order that we can

35:00 Do them some service. Some service? Yes, we were going to, but we were just waiting for you to come back. Can we come and join you? Sorry. After all that... Oh, I see where the old chairs went. It wasn't the, it wasn't their insurance, it's just that they... Two, two. Oh, you are part of the conference. Well, you can join us. Oh, sorry, we haven't got an election yet. Yeah, probably they have me. Well, I haven't elected because I bought my own. I thought you were not even in the conference. Do notes exist for hardwaters? Very much not. See, that's what I was trying to say. Well done. At least, if they're not going to, you know, meet and become our experts, if they do have notes written up, maybe they can just let people know, you know, at least have a person to look at them, because if everybody has written that up, then that's what they need all the time. One-and-a-half or two-and-a-half years, it seems like it's part of computer science, or the Church, which will be allowed to continue. But then, by then, then... I thought Gray really was trying to have it both ways. On the one hand, we're all being told that there still wasn't enough distance from the definitions to the frontier of research to yield significant difficulties to make it... ...admired in a way that number theory is admired, or real problem-solving treatments in number theory are admired, and then, you know, we'd have somebody turn around and tell us that we'd be making it all much too difficult for these poor differential geometries, much, much too difficult, and it has to be simplified and rendered, yes, it has to be rendered much more...

37:30 It's a difficult problem. It's also establishing new insights, different points of view, different questions, different perceptions, conceptions. I think someone will agree about this. It's not only just attacking conjectural problems. You know, I think of category theory very much in the way I think of, you know, the abstract algebra in the generation preceding, right, which sort of unified an awful lot of mathematics, you know, at least the way people describe it, sort of that appeared on the scene and a lot of stuff that had been extremely difficult to learn now became so easy that anybody could learn it. So, universal algebras still don't know about hyperdoctrines. They still, after 20 years, haven't bothered to read the basic paper in the subject. It's a bit as if people, I don't know, 30 years after... And yet there are even people in philosophy departments now who know about hyperdoctrines, who are doing work on possible world semantics and predicates like Peruzzi and his colleagues who use this theory. So it's disgraceful if people in mainstream, in the very developed parts of mainstream mathematics are not aware of it. And it's philosophy. All magicians are working on a lot of different problems.

40:00 It has an aspect, I don't know, mathematical. Yeah, it's science. Who is this friend of yours in the Supreme Court in Rome who is a close follower of category theory and good sort of... Seeker of the truth. I think Rosanna Khujani's talk will actually be about my work with Benjamin Fletcher. Is that the talk tomorrow evening? Is that the talk tomorrow evening in the session on Locales and Sheep Theory? Or am I thinking confusing? Rosario Cruciani, is that the person who's going to talk about sheet theory over quantiles, sheaves over quantiles? Well, that's the one I want very, very much to hear. Well, the paper's in French, isn't it? Yeah. People have wits about us. Oh, so it wasn't just my excessive weight, you know, it wasn't just the weight of the student. I'm afraid in my case, the YC don't lead the lecture. Yeah, well, the great thing about these is that they're very stable. We discovered these...

42:30 Looking around for several years is that the only place where philosophy is being done is in category theory. Yeah, that was my conclusion too. That's how we came to meet. I think I'd like to meet him very much indeed. Sounds a very interesting man. Actually, my impression of what's happened in philosophy is that it's exactly what Peter Fry was describing as happening in category theory. That the minute a branch of philosophy acquires any content, it's pushed out. See, that's not part of philosophy anymore. It's applied to some subject matter. Because after all, Newton was doing natural philosophy. And now they refuse to call that a philosophy because it's got a separate department of its own and, you know, it seems that's what happens all the time, you know, when you get serious and it really does something. I think I want to qualify the very minute clause. I think that's probably a little unfair. I think there have been, you know, there has been the occasional thinker in history, like Aristotle, who, one can't say that the minute that they formulated... An answer to the question in Aristotle's case on how to think of what he would have called categories. The animus destructus mode of being was not relative to that of any other thing, that it got pushed out. After all, logic wasn't entirely a moribund subject until... The 20th century histories of logic like to pretend that logic only came into existence in 1879. Well, I think it's true that it usually takes people in mathematics departments to provide the tools to clean up the subject area. I mean, modal logic is a very good case in point. People like Danilo and Beruzzi in Florence have actually taken the modal operators and have done a lot of work to show how they can be embedded in the correct topos and how, instead of taking this...

45:00 Extraordinary inflated ontology of possible worlds, sometimes referred to in philosophy debauches as barking mad, barking mad, running dog, modal realism, the kind David Lewis and Kripke also were quite pursued, but you could do it out of town in terms of conditions on the... There are a number of combinations on the arrows, or actually separators of pairs of arrows that take you from one context to another, you don't need to build this possible world, you don't need to have a platonistic metaphysics of possible worlds at all to do the things that the people trying to build semantics through quantum mobile are doing for you. And they're also thinking quite deeply about things like the relationship between extensionality and the various versions of the extensionality conditions taken in a topos and separability of subservients, relative uniform separability of topology. And whether there is some deep conceptual insight there connected with our notions of separability, whether we kind of assume a topological structure or the domain that we're in, which shows up in the toposanctus. I think all that's good work, but it's work which is on the borderline between philosophical logic and mathematics. Yeah, as you say, on the whole, when they actually develop the technical tools to really make precise the questions, the philosophers ask about the subject area, well, a particular subject area in... In logic, as practiced in philosophy, then it usually ends up with a subject within a generation being pushed out of philosophy altogether. I mean, there's some people who make mention of it, but I think that has to do with a lot of things. And also, Gonzalo Reyes. Mm, Gonzalo Reyes is somebody I was just thinking of. Yes, he's also written very much about these topos-theoretic models that went down. He's also very interested in linguistics, and I followed his work very closely. I admire Gonzalo. He's also, I think, he thinks very carefully of people around Asia as well.

47:30 Do you know Alberto Peruzzi, Florence, Thorne? He was actually here at the conference until yesterday, but he left yesterday afternoon. He's actually a philosopher, but he's a very competent magician. He's a great friend of John Bell's in London, or John Bell's now moved to Toronto, who's, well, by the standards of you and Bill, I mean, who is competent? I mean, by the standards of somebody in the philosophy department, he's very competent indeed in category theory. And has published quite widely papers on the philosophy of mathematics. He has written a good... There's a paper on the theory of descriptions and largely some expository papers for philosophers about Lambeck and Scott's work on identity systems, but using the insights from Sheikheretic methods and explaining just what it is in terms of... He's working at the moment on quantum logic, which is also my own particular interest, and he's a good hard thinker when it comes to foundational questions, and he tries to keep up with the development of the category theory itself, as far as any one... A man can keep up with developments. Bill told us a little last year when he was in London about the work that he and you have done. I wish I could find the time to learn more and study this.

50:00 You and Bill and Walter Mullen have written papers. Are they all sort of scattered in the journal entries and conference proceedings? Are you thinking of bringing all these ideas together in a book? Now we're trying to write a... No, just for general university students, it's pretty experimental, of course, that's what I'm saying. Well, we've never seen an introduction like that. But I think the little bit of stuff we did about this sort of combination of the intuitionistic and anti-intuitionistic sets is sort of scattered around intuition areas. Yeah, I've read parts of it. I mean, I've read the... I mean, I found it very, very difficult, so it wasn't trying some mathematicians, but you just make it, it's very difficult to count, but, you know, much tougher than reading, you know, Bill's papers, and that's saying something for a non-mathematician, but I tried. Good night. Thank you very much for your time. This is simply the latest draft of a century of similar thought that he's given on. I don't know anything about the journal. It's a philosophical journal, but maybe it was something that was sort of not written from the book.

52:30 I wish the Karazahs just showed me this article. Thank you for watching this video. I was wondering if I could come down to the subject here, but that's a lesson to be learned. Where do I go to know more stuff like that? Because I've heard that there are different ways of defining the numbers. Well, since you've done that, it's something I've done in a very minor way to show that it's a very big issue. I think it was a good time because you were trying to capture the geometrical. You can have both of them. Without these creatures being complemented, it's sort of uncomfortable. What Putnam and the others were proposing, the various properties that make you say this proposition is the negation of not A. So you can have those two. The idea of complement is, the common gate of a race proposed by the quantum of the human, does not even exist in this... Well, I don't know, it's just if one did want some kind of... If you don't want to study the 70s, it takes pretty long to study the 70s, but at least if you read a little bit of Anders Cox's book, it would give you a little bit of the flavor of the 70s, so you'd at least have an idea of what it is.

55:00 Well, I don't know about this book. No, it doesn't kind of sound interesting. What is, what is, I'm just asking as humble seekers of truth and they're out of the area, what is Penon's powerful intrinsic notion of open sub-objects? What is this, you're discussing the The sublattice of closed sub-objects has a strong distributive property as both hiding and co-hiding. You mentioned Pannon's powerful intrinsic notion of open sub-objects. Can you tell us a bit about that? Pannon is a Phrygian mathematician. Thank you for your attention. In intuitionistic predictive calculus, one can define those who are on the special orbit. There is an intrinsic property, which turns out, topological models in the intuitionistic model, nothing to do with that, sorry, something completely intrusive, some things are open, some things are not. Topology is a structure, right? It's an additional position of the structure that certain subsets are declared to be good, but by contrast, topos theory... There is intrinsic to the definition of which there's a loop, which turns out that many examples of these low topos tend to be, in spite of the fact that it's logical, not talking about additional structure, just about set theories of this type of intuition, and unless it turns out to coincide with the...

57:30 The definition is based on the idea of a... In classical logic, A implies B. It's equivalent to not A or B. But in intuitionistic logic, it's different. You could say that A is strongly contained in B if not A or B is true. Sub-object and given space. You can consider a sub-object and a given space, but you can also consider it as a verb. Various cartesian products are associated. So, essentially, if the u is contained in x, if you just define it to be open, if inside x cross x, you cross x, inside u cross x, you have the diagonal, well, you have u cross u, outside that you have the top of u, and that's the diagonal of u cross u. You say that this inclusion of the diagram in order to u-cross you inside this space or cross back is a strong inclusion, because it's not the first or the second or third, so that's a property using not just the polar oppositional logarithms of objects of x, but the fragment of x. Critic of logic, I can compare x and x cross x, but it is not an intrinsic property that you, what you might think of as a formula, is a variable, a type u or whatever, but in any case, it is the case or not the case that you lift it up in that way, satisfy this problem in your logic, in relation to the diagonal, and so you say you use open logic. So most of objects that you would analyze, you prove that...

1:00:00 We've proved that these form a hiding algebra, but not all in the sense that they're invariant under intercepted. So every map in the TOCO's profession is continuous in the sense that the inverse of one of these opens is open. That's the theorem. Now, I'll tell you the proof of that. Very interesting indeed. Say that an element of a subset of a set is well contained in the set of all the elements except for the subset Q. All wires can be connected and all wires can take a view. All y in the whole space, either y is contained in u or y is contained in x. That is usually not the case. Because the jobs are already too long. So that is what it means for x to be well contained in some set u. And the set is open if every part of it is well contained in it. I don't know why it's in the logical property, because if there's a product in here, especially an enclosed family and neighborhood. Sorry, say again. It turns out to have this logical property instead of... This well contains it. It turns out to be illogical, rather than comparing it with a classical way of expressing it. Yes, that's right, that's right. So, say again? It's neo-logicism, because the concepts come from logic, not by being imposed. It's an openness that just grows out of the logic, meaning out of the focus. Yes, but out of a beautifully geometrical way of looking at things. I mean, the traditional way of expressing the situation is that something is open if every point happens well in sighted, but in terms of this imposed family of neighborhoods, good neighborhoods, here it just becomes illogical. Where do we go for reckoning?

1:02:30 Back to his thesis or? Back to his thesis or? There is also a number of technical terms such as algebra, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, I think I can, yes, Bunga, that's Mark Bunga and Edward de Boeck, yeah, Maryland, 1980, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, I remember that, yeah, There's also, I have, the intrinsic perfectness notion, or for example, if you have an element x of a space, then in particular it's a subset, but it's a form of double negation relative to big X. So, the idea is that not not x is the germ of x. The final point is the infinitesimal germ. This again turns out to be true, that is to say, in the universe, or less true than the infinitesimal germ. In our end, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, I've got to have to go away and learn this stuff, this is absolutely fascinating.

1:05:00 What germs actually mean depends on the topos you're in. So, for example, if it's an algebraic drama, then germs mean arbitrary nilposes, more or less. Whereas in the infinitive or topological phase, germs are three terms. They represent germs in some cases. But an algebraic drama is the best version of that. The power series and the power shift correspond in the form of the power series, the form of the power series. The form of the power series themselves are precisely the functions of the series of space. The thing with space is that it's not zero. Yes, of course. Well, there's certainly deep connections for them. There are also certain ideas that we're expressing. And also, including, I would imagine... It's become much more direct. Ah, okay. Well, what was the... Inclusion, I suspect, a lot of insights about it. It's not secret. Committal. Yeah, sure. Well, basically the constraint was that for each person they allowed 23 pages, so we settled for three copies, one for me, one for the lady, one for the friend. I don't know how many they did. I didn't pick up my copy. I left either two re-copies. Thank you for your attention and see you in the next video.

1:07:30 One said that B is well-contained in the sub-set U, if that we can specialize to the occasion where B is a singleton later on, but B is less contained in U, either U or non-B, excepting in the one-point overlap. We want to co-contain U, to put together. Usually B is contained in U means that B implies U, but it's well-contained U if it strongly implies U in the classical sense. So you replace containments with, where we usually have the implications, with containments where it's wrong is the case that you find a term from the base to the junction. So that is well contained. That is well contained. Yes, and now you specialize that. I think there is well-contained between subsets, there is not some other subset for a point, then you say that point is well-contained, that's the whole subset, which means that the point of the subset is well-contained, then you say that the subset is open, and I think this implies that if an open subset is, if it contains x, it also contains the double negation closure of the point. Thank you for your attention. No, it's not a self-open open. In fact, in many cases, it's not not single-connected. It's an intersection of all the opens and containers. All the opens and the only opens. Right, so it's container every open, and it may be equal to the intersection of all opens. Yes, yes, this is very interesting.

1:10:00 Thank you for your attention. Thank you for your attention. So that's sort of what sets us back. You know, with that, I don't know the opposite of what I was going to say, but I don't know the opposite of what I was going to say. I don't know what I was going to say, but I don't know what I was going to say, but I don't know what I was going to say, but I don't know what I was going to say, but I don't know what I was going to say. Thank you for your attention. Did you ever see this movie? No, but it was a good movie. It was a good movie. It was a good movie. It was a good movie. It was a good movie. It was a good movie. It was a good movie. It was a good movie. All of these are special. Thank you for your attention.

1:12:30 So she really worked all this out. I remember the first time I taught, we started to get into the log of what's right and what's wrong. And I realized that my impression was that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. And I realized that when I was looking at them, they were just X and Y. I started to write out a table and it got to be a very large table and I was like, man, I think that is not going to work, and then you have to start to get these rules. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, mathematics, geometry theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, Thank you for your attention. One of the, in one of the rooms, not just in the media space, but in the college space, you could have the talk. Well, uh, they weren't exactly there, but they had something more or less important to do with that.

1:15:00 The concept of the paper is thought to be a kind of physics, but it's still a great technical term. But one of the main levels of study of mathematics is that it uses some of the ecus on Rawls' equationality to give us, to equify us, the data which I made an account of a few years ago, ten years ago, about the category of the Tau map here. So like if you like negative one to the one-third, you write it as two-thirds or something like that. I don't know what it should be. So, what's really going on is the fact that these three systems that I described before are three different things, and any one of which all of those right now are valid, but you're not allowed to mix these up. It's sort of a set of two things, and one of them is that if some expression has no value at all, you have no difference at all, or if you use the same type of formula, therefore there's no difference at all. And it's because, yeah, in reality, it's really difficult to do that.

1:17:30 The three systems are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60. The bases are totally arbitrary by the way they're explained in the natural document file. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Well, sometimes they just present it in one piece. Thank you for your attention. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, mathematics, physics, and mathematics. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, mathematics, and mathematics. Thank you for your attention. I don't really know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. I don't know why you put your name on the blackboard. Thank you for your attention. I put in one-third of what I knew, so I made a proof of that, and I actually counted it as a negative 1.

1:20:00 I could take the 49th, but I couldn't take the 7th, so I'm going to take the 13th. So if you've ever played with the algebra, I don't know if you've ever played with the algebra, I don't know if you've ever played with the algebra, I don't know if you've ever played with the algebra, I don't know if you've ever played with the algebra, I don't know if you've ever played with the algebra, I don't know if you've Thank you for your attention. We can make a copy of it in the morning. We may have to physically show you and say 20 things. You know what I mean? Maybe he has another use for it. I mean, if you want to use it, otherwise they begin to challenge you. Yeah? Well, you get it done at the building itself. I don't want to use it at that time. No, you, well, I won't have time for it tonight. I'm just a bit too, I don't know. And it's... There's two developments for me, where I follow the ideas of, well, this is beautiful. Well, this, I think, well, let's try an example. I don't think it's going to work like this. I would propose it's going to be like the thing that I'm coming up with. I think it's a good... Some name like that, but it's not... I would call it a navigation. But I don't think it's going to work like this. Well, we have to get back to the hotel at 12.18. We have to walk back there. 12.18 up to 12.25. We have to be part of it.

1:22:30 The correct convention is 0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0- If you have a family of open sets, then one way to define the pre-sheet is to say, let's take the continuous functions of A to R, and that's a pre-sheet, but another way to define a pre-sheet is to say, let's take the real value functions of A to R, and that's a pre-sheet, but another way to define a pre-sheet is to say, let's take the real value functions of A to R, There is no longer a sheaf for the use of text, so it has a two-location, and this is what has to go out of the process of reading the elements that go to our curriculum, and the process of reading is to use the culture of the future. The functions on the N plus 1 are equal near the diagonal. There's a sheath, which is not a function, but it's an important construction for a particular case in case. Thank you for your attention. There is an equivalence relation between the two jobs, and I appreciate that if you keep it by it, it's no longer an equivalence relation, because the law of transitivity is in there, but it's still an object.

1:25:00 We want to give a name to Tilda A. Tilda A. We want to call it Neglection. Neglection of the hating one. Negation. The other one is Neglection. Now I'm not making a pun there. I'd rather call this one Negation because it's dialectical. This one is Negation. I told you it is. Because one is based on... One is based on contradictions in the sense that you end it with A, you get zero, while the other one you over it with A and you get one, meaning this one is completable. It's the smallest thing that completes A. So it is what A has neglected to say and now it completes it. While the other one what A purposely doesn't want you to say because you've contradicted negation and this one is completion to some extent. I would have almost called it complement because it completes it. I mean, I don't know. I don't know. Well, you could almost want to say that. It does capture that notion of what A has neglected to say. I mean, what A needs to say and what the complete A needs to say. It's what comes uncompleted. One what the other one.